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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b4">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font> ex <font color="Olive" title="b5">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b4">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> );<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ;<br/>
<div><span class="lab"><font color="Green" title="E47">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z1</font>, <font color="Maroon" title="c6">z2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c6">z2</font>] implies <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:58_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)
 <span class="kw">and </span><br/><a NAME="E2:58_1_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c6">z2</font>] implies <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font> )</span><br/><span class="kw">assume </span><a NAME="E3:58_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z1</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c6">z2</font>] implies <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">F</font>, <font color="Maroon" title="c8">F1</font>, <font color="Maroon" title="c9">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c10">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">F1</font>, <font color="Maroon" title="c11">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c8">F1</font>,<font color="Maroon" title="c9">F2</font><span class="kw"> such that </span><br/><a NAME="E5:58_1_1_1"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c10">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:58_1_1_1"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F1</font>,<font color="Maroon" title="c9">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:58_1_1_1"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c9">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c10">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><span class="kw">assume </span><a NAME="E8:58_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c6">z2</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">F9</font>, <font color="Maroon" title="c13">F19</font>, <font color="Maroon" title="c14">F29</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c15">t9</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c12">F9</font>,<font color="Maroon" title="c13">F19</font>, <font color="Maroon" title="c16">t19</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F19</font>,<font color="Maroon" title="c14">F29</font><span class="kw"> such that </span><br/><a NAME="E10:58_1_1_1"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F9</font>,<font color="Maroon" title="c13">F19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">t9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E11:58_1_1_1"/><span class="lab"><font color="Green" title="E52">A6</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F19</font>,<font color="Maroon" title="c14">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c16">t19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E12:58_1_1_1"/><span class="lab"><font color="Green" title="E53">A7</font></span>: 
<font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F9</font>,<font color="Maroon" title="c14">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">t19</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c15">t9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><a NAME="E13:58_1_1_1"/><span class="lab"><font color="Green" title="E54">A8</font></span>: 
<font color="Maroon" title="c10">t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c15">t9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E10:58_1_1_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E14:58_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F1</font>,<font color="Maroon" title="c9">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F19</font>,<font color="Maroon" title="c14">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E11:58_1_1_1"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E15:58_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A9</font></span>: 
<font color="Maroon" title="c9">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c14">F29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E16:58_1_1_1"/><span class="lab"><font color="Green" title="E56">A10</font></span>: 
<font color="Maroon" title="c11">t1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c16">t19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E11:58_1_1_1"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E17:58_1_1_1"/><span class="lab"><font color="Green" title="E57">A11</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F9</font>,<font color="Maroon" title="c13">F19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E10:58_1_1_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E18:58_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c12">F9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E19:58_1_1_1"/>
<font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="txt" href="#E12:58_1_1_1"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="txt" href="#E17:58_1_1_1"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="txt" href="#E13:58_1_1_1"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E16:58_1_1_1"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="txt" href="#E15:58_1_1_1"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E48">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font>] implies <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:58_1_1_2"/><span class="lab"><font color="Green" title="E49">A13</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)
 <span class="kw">and </span><br/><a NAME="E2:58_1_1_2"/><span class="lab"><font color="Green" title="E50">A14</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font>] implies <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) )</span><br/><span class="kw">assume </span><a NAME="E3:58_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">F</font>, <font color="Maroon" title="c7">F1</font>, <font color="Maroon" title="c8">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c9">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">F1</font>, <font color="Maroon" title="c10">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c8">F2</font><span class="kw"> such that </span><br/><a NAME="E5:58_1_1_2"/><span class="lab"><font color="Green" title="E51">A15</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c9">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:58_1_1_2"/><span class="lab"><font color="Green" title="E52">A16</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c8">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c10">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:58_1_1_2"/><span class="lab"><font color="Green" title="E53">A17</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c8">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c9">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><a NAME="E8:58_1_1_2"/><span class="lab"><font color="Green" title="E54">A18</font></span>: 
<font color="Maroon" title="c7">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_2"><span class="lab"><font color="Green" title="E49">A13</font></span></a>, <a class="txt" href="#E6:58_1_1_2"><span class="lab"><font color="Green" title="E52">A16</font></span></a>, <a class="ref" href="nattra_1.html#T32" target="_self" title="NATTRA_1:th.32">Th35</a></span>;<br/><a NAME="E9:58_1_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_2"><span class="lab"><font color="Green" title="E50">A14</font></span></a>, <a class="txt" href="#E5:58_1_1_2"><span class="lab"><font color="Green" title="E51">A15</font></span></a>, <a class="ref" href="nattra_1.html#T32" target="_self" title="NATTRA_1:th.32">Th35</a></span>;<br/><a NAME="E10:58_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">F</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1_2"><span class="lab"><font color="Green" title="E54">A18</font></span></a>, <a class="ref" href="nattra_1.html#T23" target="_self" title="NATTRA_1:th.23">Th25</a></span>;<br/><span class="kw">hence </span><a NAME="E11:58_1_1_2"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1_2"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="ref" href="nattra_1.html#T32" target="_self" title="NATTRA_1:th.32">Th35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c3">m</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:58_1_1"/><span class="lab"><font color="Green" title="E49">A19</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">m</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) &amp;  ex <font color="Olive" title="b3">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] ) )
 <span class="kw">and </span><br/><a NAME="E5:58_1_1"/><span class="lab"><font color="Green" title="E50">A20</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">m</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c3">m</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S5" title="BINOP_1:sch.5">BINOP_1:sch 5</a>(<a class="txt" href="#E2:58_1_1"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="txt" href="#E1:58_1_1"><span class="lab"><font color="Green" title="E47">A1</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E51">A21</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">t</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">t</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span><br/><span class="kw">consider </span><font color="Maroon" title="c5">F1</font>, <font color="Maroon" title="c6">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c7">s</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font><span class="kw"> such that </span><br/><a NAME="E2:58_1_1_3"/><span class="lab"><font color="Green" title="E52">A22</font></span>: 
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:58_1_1_3"/>
<font color="Maroon" title="c5">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_3"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c5">F1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:58_1_1_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">t</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#D2" title="CAT_2:def.2">CAT_2:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c4">d</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E8:58_1_1"/><span class="lab"><font color="Green" title="E52">A23</font></span>: 
 for <font color="Olive" title="b1">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) holds  <font color="Maroon" title="c4">d</font> <a href="cat_2.html#K6" title="CAT_2:func.6">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">t</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S8" title="FUNCT_2:sch.8">FUNCT_2:sch 8</a>(<a class="txt" href="#E6:58_1_1"><span class="lab"><font color="Green" title="E51">A21</font></span></a>);<br/></span>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ;<br/>
<div><span class="lab"><font color="Green" title="E53">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">t</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">t</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span><br/><span class="kw">consider </span><font color="Maroon" title="c6">F1</font>, <font color="Maroon" title="c7">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c8">s</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font><span class="kw"> such that </span><br/><a NAME="E2:58_1_1_4"/><span class="lab"><font color="Green" title="E54">A25</font></span>: 
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c8">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:58_1_1_4"/>
<font color="Maroon" title="c6">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_4"><span class="lab"><font color="Green" title="E54">A25</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c7">F2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:58_1_1_4"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">t</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#D2" title="CAT_2:def.2">CAT_2:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c5">c</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E11:58_1_1"/><span class="lab"><font color="Green" title="E54">A26</font></span>: 
 for <font color="Olive" title="b1">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) holds  <font color="Maroon" title="c5">c</font> <a href="cat_2.html#K6" title="CAT_2:func.6">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">t</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S8" title="FUNCT_2:sch.8">FUNCT_2:sch 8</a>(<a class="txt" href="#E9:58_1_1"><span class="lab"><font color="Green" title="E53">A24</font></span></a>);<br/></span>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="cat_2.html#M2" title="CAT_2:mode.2">Element</a> of  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,$1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> $1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E12:58_1_1"/><span class="lab"><font color="Green" title="E55">A27</font></span>: 
 for <font color="Olive" title="b1">F</font> being    <a href="cat_2.html#M2" title="CAT_2:mode.2">Element</a> of  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) holds  <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">F</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">i</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E14:58_1_1"/><span class="lab"><font color="Green" title="E56">A28</font></span>: 
 for <font color="Olive" title="b1">F</font> being    <a href="cat_2.html#M2" title="CAT_2:mode.2">Element</a> of  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) holds  <font color="Maroon" title="c6">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">F</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S8" title="FUNCT_2:sch.8">FUNCT_2:sch 8</a>(<a class="txt" href="#E12:58_1_1"><span class="lab"><font color="Green" title="E55">A27</font></span></a>);<br/></span>
<span class="kw">set </span><font color="Maroon" title="c7">C</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #);<br/>
<div><span class="lab"><font color="Green" title="E57">A29</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">F</font>, <font color="Maroon" title="c9">F1</font>, <font color="Maroon" title="c10">F2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font><br/>  for <font color="Olive" title="b2">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font>  st <font color="Maroon" title="c9">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c10">F2</font> &amp; <font color="Maroon" title="c8">F</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c9">F1</font> holds <br/><font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b2">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b1">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b1">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c11">t</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font>  st <font color="Maroon" title="c9">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c10">F2</font> &amp; <font color="Maroon" title="c8">F</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c9">F1</font> holds <br/><font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b1">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c11">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c12">t1</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c10">F2</font> &amp; <font color="Maroon" title="c8">F</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c9">F1</font> implies <font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c11">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:58_1_1_5"/><span class="lab"><font color="Green" title="E58">A30</font></span>: 
<font color="Maroon" title="c9">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c10">F2</font>
 <span class="kw">and </span><br/><a NAME="E2:58_1_1_5"/><span class="lab"><font color="Green" title="E59">A31</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c9">F1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c11">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/><a NAME="E3:58_1_1_5"/><span class="lab"><font color="Green" title="E60">A32</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_5"><span class="lab"><font color="Green" title="E59">A31</font></span></a>, <a class="ref" href="nattra_1.html#T32" target="_self" title="NATTRA_1:th.32">Th35</a></span>;<br/><a NAME="E4:58_1_1_5"/><span class="lab"><font color="Green" title="E61">A33</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c11">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]
 ;<br/><a NAME="E5:58_1_1_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_5"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="ref" href="nattra_1.html#T32" target="_self" title="NATTRA_1:th.32">Th35</a></span>;<br/><a NAME="E6:58_1_1_5"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1"><span class="lab"><font color="Green" title="E49">A19</font></span></a>, <a class="txt" href="#E3:58_1_1_5"><span class="lab"><font color="Green" title="E60">A32</font></span></a>, <a class="txt" href="#E4:58_1_1_5"><span class="lab"><font color="Green" title="E61">A33</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">F9</font>, <font color="Maroon" title="c14">F19</font>, <font color="Maroon" title="c15">F29</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c16">t9</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F9</font>,<font color="Maroon" title="c14">F19</font>, <font color="Maroon" title="c17">t19</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c14">F19</font>,<font color="Maroon" title="c15">F29</font><span class="kw"> such that </span><br/><a NAME="E8:58_1_1_5"/><span class="lab"><font color="Green" title="E62">A34</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F9</font>,<font color="Maroon" title="c14">F19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c16">t9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:58_1_1_5"/><span class="lab"><font color="Green" title="E63">A35</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F19</font>,<font color="Maroon" title="c15">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">t19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:58_1_1_5"/><span class="lab"><font color="Green" title="E64">A36</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F9</font>,<font color="Maroon" title="c15">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">t19</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c16">t9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E50">A20</font></span></a></span>;<br/><a NAME="E11:58_1_1_5"/><span class="lab"><font color="Green" title="E65">A37</font></span>: 
<font color="Maroon" title="c11">t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c16">t9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1_5"><span class="lab"><font color="Green" title="E62">A34</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E12:58_1_1_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F19</font>,<font color="Maroon" title="c15">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:58_1_1_5"><span class="lab"><font color="Green" title="E63">A35</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E13:58_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A38</font></span>: 
<font color="Maroon" title="c10">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c15">F29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E14:58_1_1_5"/><span class="lab"><font color="Green" title="E67">A39</font></span>: 
<font color="Maroon" title="c12">t1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c17">t19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:58_1_1_5"><span class="lab"><font color="Green" title="E63">A35</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E15:58_1_1_5"/><span class="lab"><font color="Green" title="E68">A40</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F9</font>,<font color="Maroon" title="c14">F19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1_5"><span class="lab"><font color="Green" title="E62">A34</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E16:58_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c8">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">F9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E17:58_1_1_5"/>
<font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c11">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:58_1_1_5"><span class="lab"><font color="Green" title="E64">A36</font></span></a>, <a class="txt" href="#E15:58_1_1_5"><span class="lab"><font color="Green" title="E68">A40</font></span></a>, <a class="txt" href="#E11:58_1_1_5"><span class="lab"><font color="Green" title="E65">A37</font></span></a>, <a class="txt" href="#E14:58_1_1_5"><span class="lab"><font color="Green" title="E67">A39</font></span></a>, <a class="txt" href="#E13:58_1_1_5"><span class="lab"><font color="Green" title="E66">A38</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">thus </span><a NAME="E1:58_1_1_6"/><span class="lab"><font color="Green" title="E58">A41</font></span>: 
 for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) iff  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/>(  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> ) ) &amp; (  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> &amp;  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">h</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">g</font>)</span>)</span>,<font color="Olive" title="b1">f</font>) ) &amp; (  for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) holds <br/> (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; (  for <font color="Olive" title="b2">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span>,<font color="Olive" title="b2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) &amp; (  for <font color="Olive" title="b2">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">g</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) ) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) iff  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">F19</font>, <font color="Maroon" title="c11">F29</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c12">t9</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">F19</font>,<font color="Maroon" title="c11">F29</font><span class="kw"> such that </span><br/><a NAME="E2:58_1_1_6_1"/><span class="lab"><font color="Green" title="E59">A42</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F19</font>,<font color="Maroon" title="c11">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:58_1_1_6_1"/><span class="lab"><font color="Green" title="E60">A43</font></span>: 
<font color="Maroon" title="c10">F19</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">F29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:58_1_1_6_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) implies  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:58_1_1_6_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">F</font>, <font color="Maroon" title="c14">F1</font>, <font color="Maroon" title="c15">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c16">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F</font>,<font color="Maroon" title="c14">F1</font>, <font color="Maroon" title="c17">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c15">F2</font><span class="kw"> such that </span><br/><a NAME="E3:58_1_1_6_1_1"/><span class="lab"><font color="Green" title="E61">A44</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F</font>,<font color="Maroon" title="c14">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c16">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:58_1_1_6_1_1"/><span class="lab"><font color="Green" title="E62">A45</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c15">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:58_1_1_6_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F</font>,<font color="Maroon" title="c15">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c16">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E50">A20</font></span></a></span>;<br/>
<span class="kw">thus </span> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c15">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_6_1_1"><span class="lab"><font color="Green" title="E62">A45</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c14">F1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F</font>,<font color="Maroon" title="c14">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1_6_1_1"><span class="lab"><font color="Green" title="E61">A44</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1"><span class="lab"><font color="Green" title="E54">A26</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E5:58_1_1_6_1"/><span class="lab"><font color="Green" title="E61">A46</font></span>: 
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c13">F1</font>, <font color="Maroon" title="c14">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c15">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font><span class="kw"> such that </span><br/><a NAME="E7:58_1_1_6_1"/><span class="lab"><font color="Green" title="E62">A47</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:58_1_1_6_1"/><span class="lab"><font color="Green" title="E63">A48</font></span>: 
<font color="Maroon" title="c13">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c14">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></span>;<br/>
<span class="lab"><font color="Green" title="E64">A49</font></span>: <font color="Maroon" title="c13">F1</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1_6_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c5">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="txt" href="#E5:58_1_1_6_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1"><span class="lab"><font color="Green" title="E54">A26</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F19</font>,<font color="Maroon" title="c11">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_6_1"><span class="lab"><font color="Green" title="E59">A42</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c11">F29</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">t</font> = <font color="Maroon" title="c15">t</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c11">F29</font>,<font color="Maroon" title="c14">F2</font> ;<br/>
<a NAME="E11:58_1_1_6_1"/>
<font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F19</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c12">t9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:58_1_1"><span class="lab"><font color="Green" title="E57">A29</font></span></a>, <a class="txt" href="#E7:58_1_1_6_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E8:58_1_1_6_1"><span class="lab"><font color="Green" title="E63">A48</font></span></a>, <a class="txt" href="#E2:58_1_1_6_1"><span class="lab"><font color="Green" title="E59">A42</font></span></a>, <a class="txt" href="#E3:58_1_1_6_1"><span class="lab"><font color="Green" title="E60">A43</font></span></a>, <a class="txt" href="#E9:58_1_1_6_1"><span class="lab"><font color="Green" title="E64">A49</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:58_1_1_6_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1"><span class="lab"><font color="Green" title="E49">A19</font></span></a>, <a class="txt" href="#E7:58_1_1_6_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E2:58_1_1_6_1"><span class="lab"><font color="Green" title="E59">A42</font></span></a>, <a class="txt" href="#E9:58_1_1_6_1"><span class="lab"><font color="Green" title="E64">A49</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E2:58_1_1_6"/>
 for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/>(  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> &amp;  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">h</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">g</font>)</span>)</span>,<font color="Olive" title="b1">f</font>) ) &amp; (  for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) holds <br/> (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; (  for <font color="Olive" title="b2">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span>,<font color="Olive" title="b2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) &amp; (  for <font color="Olive" title="b2">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">g</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) ) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> implies (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">F19</font>, <font color="Maroon" title="c11">F29</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c12">t9</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">F19</font>,<font color="Maroon" title="c11">F29</font><span class="kw"> such that </span><br/><a NAME="E2:58_1_1_6_2"/><span class="lab"><font color="Green" title="E59">A50</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F19</font>,<font color="Maroon" title="c11">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:58_1_1_6_2"/><span class="lab"><font color="Green" title="E60">A51</font></span>: 
<font color="Maroon" title="c10">F19</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">F29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:58_1_1_6_2"/>
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> )</span><br/>

<a NAME="E5:58_1_1_6_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_6"><span class="lab"><font color="Green" title="E58">A41</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">F</font>, <font color="Maroon" title="c14">F1</font>, <font color="Maroon" title="c15">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c16">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F</font>,<font color="Maroon" title="c14">F1</font>, <font color="Maroon" title="c17">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c15">F2</font><span class="kw"> such that </span><br/><a NAME="E7:58_1_1_6_2"/><span class="lab"><font color="Green" title="E61">A52</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F</font>,<font color="Maroon" title="c14">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c16">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:58_1_1_6_2"/><span class="lab"><font color="Green" title="E62">A53</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c15">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:58_1_1_6_2"/><span class="lab"><font color="Green" title="E63">A54</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F</font>,<font color="Maroon" title="c15">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c16">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E50">A20</font></span></a></span>;<br/>
<a NAME="E10:58_1_1_6_2"/><span class="lab"><font color="Green" title="E64">A55</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F</font>,<font color="Maroon" title="c14">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F19</font>,<font color="Maroon" title="c11">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1_6_2"><span class="lab"><font color="Green" title="E61">A52</font></span></a>, <a class="txt" href="#E2:58_1_1_6_2"><span class="lab"><font color="Green" title="E59">A50</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E11:58_1_1_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A56</font></span>: 
<font color="Maroon" title="c13">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c10">F19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E12:58_1_1_6_2"/><span class="lab"><font color="Green" title="E66">A57</font></span>: 
<font color="Maroon" title="c14">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">F29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:58_1_1_6_2"><span class="lab"><font color="Green" title="E64">A55</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">F19</font>, <font color="Maroon" title="c19">F29</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c20">t9</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c18">F19</font>,<font color="Maroon" title="c19">F29</font><span class="kw"> such that </span><br/><a NAME="E14:58_1_1_6_2"/><span class="lab"><font color="Green" title="E67">A58</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">F19</font>,<font color="Maroon" title="c19">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c20">t9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E15:58_1_1_6_2"/><span class="lab"><font color="Green" title="E68">A59</font></span>: 
<font color="Maroon" title="c18">F19</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c19">F29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></span>;<br/>
<a NAME="E16:58_1_1_6_2"/><span class="lab"><font color="Green" title="E69">A60</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c15">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">F19</font>,<font color="Maroon" title="c19">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1_6_2"><span class="lab"><font color="Green" title="E62">A53</font></span></a>, <a class="txt" href="#E14:58_1_1_6_2"><span class="lab"><font color="Green" title="E67">A58</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E17:58_1_1_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A61</font></span>: 
<font color="Maroon" title="c15">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c19">F29</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E18:58_1_1_6_2"/>
<font color="Maroon" title="c14">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c18">F19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:58_1_1_6_2"><span class="lab"><font color="Green" title="E69">A60</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E19:58_1_1_6_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">F</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c15">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1_6_2"><span class="lab"><font color="Green" title="E60">A51</font></span></a>, <a class="txt" href="#E11:58_1_1_6_2"><span class="lab"><font color="Green" title="E65">A56</font></span></a>, <a class="txt" href="#E12:58_1_1_6_2"><span class="lab"><font color="Green" title="E66">A57</font></span></a>, <a class="txt" href="#E15:58_1_1_6_2"><span class="lab"><font color="Green" title="E68">A59</font></span></a>, <a class="txt" href="#E17:58_1_1_6_2"><span class="lab"><font color="Green" title="E70">A61</font></span></a>, <a class="ref" href="nattra_1.html#T23" target="_self" title="NATTRA_1:th.23">Th25</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F</font>,<font color="Maroon" title="c15">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c16">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#T32" target="_self" title="NATTRA_1:th.32">Th35</a></span>;<br/>
<span class="kw">thus </span> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font>)</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="txt" href="#E9:58_1_1_6_2"><span class="lab"><font color="Green" title="E63">A54</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F</font>,<font color="Maroon" title="c15">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c13">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F</font>,<font color="Maroon" title="c14">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F</font>,<font color="Maroon" title="c14">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c16">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="txt" href="#E7:58_1_1_6_2"><span class="lab"><font color="Green" title="E61">A52</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font></span><br/>

<span class="kw">thus </span> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font>)</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E9:58_1_1_6_2"><span class="lab"><font color="Green" title="E63">A54</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F</font>,<font color="Maroon" title="c15">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c15">F2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c15">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c15">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E8:58_1_1_6_2"><span class="lab"><font color="Green" title="E62">A53</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E3:58_1_1_6"/>
 for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> &amp;  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">h</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">g</font>)</span>)</span>,<font color="Olive" title="b1">f</font>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) holds <br/> (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; (  for <font color="Olive" title="b2">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span>,<font color="Olive" title="b2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) &amp; (  for <font color="Olive" title="b2">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">g</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font>, <font color="Maroon" title="c10">h</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> &amp;  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> implies  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c10">h</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c10">h</font>,<font color="Maroon" title="c9">g</font>)</span>)</span>,<font color="Maroon" title="c8">f</font>) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">f9</font> = <font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c12">g9</font> = <font color="Maroon" title="c9">g</font>, <font color="Maroon" title="c13">h9</font> = <font color="Maroon" title="c10">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) ;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:58_1_1_6_3"/><span class="lab"><font color="Green" title="E59">A62</font></span>: 
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font>
 <span class="kw">and </span><br/><a NAME="E3:58_1_1_6_3"/><span class="lab"><font color="Green" title="E60">A63</font></span>: 
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c10">h</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c10">h</font>,<font color="Maroon" title="c9">g</font>)</span>)</span>,<font color="Maroon" title="c8">f</font>)</span><br/>

<a NAME="E4:58_1_1_6_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">g9</font>,<font color="Maroon" title="c11">f9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_6"><span class="lab"><font color="Green" title="E58">A41</font></span></a>, <a class="txt" href="#E3:58_1_1_6_3"><span class="lab"><font color="Green" title="E60">A63</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">F1</font>, <font color="Maroon" title="c15">F19</font>, <font color="Maroon" title="c16">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c17">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c15">F19</font>, <font color="Maroon" title="c18">t2</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c15">F19</font>,<font color="Maroon" title="c16">F2</font><span class="kw"> such that </span><br/><a NAME="E6:58_1_1_6_3"/><span class="lab"><font color="Green" title="E61">A64</font></span>: 
<font color="Maroon" title="c11">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c15">F19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:58_1_1_6_3"/><span class="lab"><font color="Green" title="E62">A65</font></span>: 
<font color="Maroon" title="c12">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F19</font>,<font color="Maroon" title="c16">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c18">t2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:58_1_1_6_3"/><span class="lab"><font color="Green" title="E63">A66</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c12">g9</font>,<font color="Maroon" title="c11">f9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c16">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">t2</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c17">t1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E50">A20</font></span></a></span>;<br/>
<a NAME="E9:58_1_1_6_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">h9</font>,<font color="Maroon" title="c12">g9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_6"><span class="lab"><font color="Green" title="E58">A41</font></span></a>, <a class="txt" href="#E2:58_1_1_6_3"><span class="lab"><font color="Green" title="E59">A62</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">F29</font>, <font color="Maroon" title="c20">F3</font>, <font color="Maroon" title="c21">F39</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c22">t29</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c19">F29</font>,<font color="Maroon" title="c20">F3</font>, <font color="Maroon" title="c23">t3</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c20">F3</font>,<font color="Maroon" title="c21">F39</font><span class="kw"> such that </span><br/><a NAME="E11:58_1_1_6_3"/><span class="lab"><font color="Green" title="E64">A67</font></span>: 
<font color="Maroon" title="c12">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F29</font>,<font color="Maroon" title="c20">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c22">t29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E12:58_1_1_6_3"/><span class="lab"><font color="Green" title="E65">A68</font></span>: 
<font color="Maroon" title="c13">h9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">F3</font>,<font color="Maroon" title="c21">F39</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c23">t3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E13:58_1_1_6_3"/>
<font color="Maroon" title="c3">m</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">h9</font>,<font color="Maroon" title="c12">g9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F29</font>,<font color="Maroon" title="c21">F39</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">t3</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c22">t29</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E50">A20</font></span></a></span>;<br/>
<a NAME="E14:58_1_1_6_3"/><span class="lab"><font color="Green" title="E66">A69</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F29</font>,<font color="Maroon" title="c20">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F19</font>,<font color="Maroon" title="c16">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1_6_3"><span class="lab"><font color="Green" title="E62">A65</font></span></a>, <a class="txt" href="#E11:58_1_1_6_3"><span class="lab"><font color="Green" title="E64">A67</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E15:58_1_1_6_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A70</font></span>: 
<font color="Maroon" title="c12">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F19</font>,<font color="Maroon" title="c20">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c18">t2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1_6_3"><span class="lab"><font color="Green" title="E62">A65</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">t2</font> = <font color="Maroon" title="c18">t2</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c15">F19</font>,<font color="Maroon" title="c20">F3</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58_1_1_6_3"><span class="lab"><font color="Green" title="E66">A69</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E17:58_1_1_6_3"/><span class="lab"><font color="Green" title="E68">A71</font></span>: 
<font color="Maroon" title="c16">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c20">F3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58_1_1_6_3"><span class="lab"><font color="Green" title="E66">A69</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E18:58_1_1_6_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A72</font></span>: 
<font color="Maroon" title="c15">F19</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c20">F3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1_6_3"><span class="lab"><font color="Green" title="E62">A65</font></span></a>, <a class="ref" href="nattra_1.html#T32" target="_self" title="NATTRA_1:th.32">Th35</a></span>;<br/>
<a NAME="E19:58_1_1_6_3"/><span class="lab"><font color="Green" title="E70">A73</font></span>: 
<font color="Maroon" title="c20">F3</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c21">F39</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:58_1_1_6_3"><span class="lab"><font color="Green" title="E65">A68</font></span></a>, <a class="ref" href="nattra_1.html#T32" target="_self" title="NATTRA_1:th.32">Th35</a></span>;<br/>
<a NAME="E20:58_1_1_6_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A74</font></span>: 
<font color="Maroon" title="c15">F19</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c21">F39</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:58_1_1_6_3"><span class="lab"><font color="Green" title="E69">A72</font></span></a>, <a class="ref" href="nattra_1.html#T23" target="_self" title="NATTRA_1:th.23">Th25</a></span>;<br/>
<a NAME="E21:58_1_1_6_3"/><span class="lab"><font color="Green" title="E72">A75</font></span>: 
<font color="Maroon" title="c14">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c15">F19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1_1_6_3"><span class="lab"><font color="Green" title="E61">A64</font></span></a>, <a class="ref" href="nattra_1.html#T32" target="_self" title="NATTRA_1:th.32">Th35</a></span>;<br/>
<a NAME="E22:58_1_1_6_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c20">F3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:58_1_1_6_3"><span class="lab"><font color="Green" title="E69">A72</font></span></a>, <a class="ref" href="nattra_1.html#T23" target="_self" title="NATTRA_1:th.23">Th25</a></span>;<br/>
<span class="kw">hence </span> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c10">h</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>) = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c21">F39</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">t3</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">t2</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c17">t1</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:58_1_1"><span class="lab"><font color="Green" title="E57">A29</font></span></a>, <a class="txt" href="#E8:58_1_1_6_3"><span class="lab"><font color="Green" title="E63">A66</font></span></a>, <a class="txt" href="#E12:58_1_1_6_3"><span class="lab"><font color="Green" title="E65">A68</font></span></a>, <a class="txt" href="#E17:58_1_1_6_3"><span class="lab"><font color="Green" title="E68">A71</font></span></a>, <a class="txt" href="#E19:58_1_1_6_3"><span class="lab"><font color="Green" title="E70">A73</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c21">F39</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">t3</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c24">t2</font></span>)</span> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c17">t1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:58_1_1_6_3"><span class="lab"><font color="Green" title="E72">A75</font></span></a>, <a class="txt" href="#E18:58_1_1_6_3"><span class="lab"><font color="Green" title="E69">A72</font></span></a>, <a class="txt" href="#E19:58_1_1_6_3"><span class="lab"><font color="Green" title="E70">A73</font></span></a>, <a class="ref" href="nattra_1.html#T26" target="_self" title="NATTRA_1:th.26">Th28</a></span>
<br/>.= 
<font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F19</font>,<font color="Maroon" title="c21">F39</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Maroon" title="c23">t3</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c24">t2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">f9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:58_1_1"><span class="lab"><font color="Green" title="E57">A29</font></span></a>, <a class="txt" href="#E6:58_1_1_6_3"><span class="lab"><font color="Green" title="E61">A64</font></span></a>, <a class="txt" href="#E21:58_1_1_6_3"><span class="lab"><font color="Green" title="E72">A75</font></span></a>, <a class="txt" href="#E20:58_1_1_6_3"><span class="lab"><font color="Green" title="E71">A74</font></span></a></span>
<br/>.= 
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c10">h</font>,<font color="Maroon" title="c9">g</font>)</span>)</span>,<font color="Maroon" title="c8">f</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:58_1_1"><span class="lab"><font color="Green" title="E57">A29</font></span></a>, <a class="txt" href="#E12:58_1_1_6_3"><span class="lab"><font color="Green" title="E65">A68</font></span></a>, <a class="txt" href="#E15:58_1_1_6_3"><span class="lab"><font color="Green" title="E67">A70</font></span></a>, <a class="txt" href="#E18:58_1_1_6_3"><span class="lab"><font color="Green" title="E69">A72</font></span></a>, <a class="txt" href="#E19:58_1_1_6_3"><span class="lab"><font color="Green" title="E70">A73</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> &amp; (  for <font color="Olive" title="b1">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> ) &amp; (  for <font color="Olive" title="b1">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">g</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c8">b</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#D2" title="CAT_2:def.2">CAT_2:def 2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">s</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c9">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></span>;<br/><a NAME="E6:58_1_1_6"/><span class="lab"><font color="Green" title="E59">A76</font></span>: 
 the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c9">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58_1_1"><span class="lab"><font color="Green" title="E56">A28</font></span></a></span>;<br/><span class="kw">hence </span> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c8">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> &amp; (  for <font color="Olive" title="b1">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> ) &amp; (  for <font color="Olive" title="b1">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">g</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) )</span><br/><span class="kw">thus </span> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E6:58_1_1_6"><span class="lab"><font color="Green" title="E59">A76</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c8">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> ) &amp; (  for <font color="Olive" title="b1">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">g</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) )</span><br/><span class="kw">thus </span><a NAME="E9:58_1_1_6"/>
 for <font color="Olive" title="b1">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #)  st  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> holds <br/> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">g</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">f</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> implies  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<font color="Maroon" title="c11">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">f9</font> = <font color="Maroon" title="c11">f</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) ;<br/>
<span class="kw">assume </span><a NAME="E2:58_1_1_6_6"/><span class="lab"><font color="Green" title="E60">A77</font></span>: 
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<font color="Maroon" title="c11">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c13">F1</font>, <font color="Maroon" title="c14">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c15">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font><span class="kw"> such that </span><br/><a NAME="E4:58_1_1_6_6"/><span class="lab"><font color="Green" title="E61">A78</font></span>: 
<font color="Maroon" title="c12">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:58_1_1_6_6"/>
<font color="Maroon" title="c13">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c14">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></span>;<br/>
<span class="lab"><font color="Green" title="E62">A79</font></span>: <font color="Maroon" title="c14">F2</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f9</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_6_6"><span class="lab"><font color="Green" title="E61">A78</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c9">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E2:58_1_1_6_6"><span class="lab"><font color="Green" title="E60">A77</font></span></a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">t</font> = <font color="Maroon" title="c15">t</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c9">F</font> ;<br/>
<a NAME="E8:58_1_1_6_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">f9</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c16">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_6_6"><span class="lab"><font color="Green" title="E61">A78</font></span></a>, <a class="txt" href="#E6:58_1_1_6_6"><span class="lab"><font color="Green" title="E62">A79</font></span></a></span>;<br/>
<a NAME="E9:58_1_1_6_6"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">f9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1"><span class="lab"><font color="Green" title="E49">A19</font></span></a></span>;<br/>
<a NAME="E10:58_1_1_6_6"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">F</font></span>)</span>,<font color="Maroon" title="c12">f9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58_1_1"><span class="lab"><font color="Green" title="E56">A28</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">F9</font>, <font color="Maroon" title="c18">F19</font>, <font color="Maroon" title="c19">F29</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c20">t9</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c17">F9</font>,<font color="Maroon" title="c18">F19</font>, <font color="Maroon" title="c21">t19</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c18">F19</font>,<font color="Maroon" title="c19">F29</font><span class="kw"> such that </span><br/><a NAME="E12:58_1_1_6_6"/><span class="lab"><font color="Green" title="E63">A80</font></span>: 
<font color="Maroon" title="c12">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F9</font>,<font color="Maroon" title="c18">F19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c20">t9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E13:58_1_1_6_6"/><span class="lab"><font color="Green" title="E64">A81</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">F19</font>,<font color="Maroon" title="c19">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c21">t19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E14:58_1_1_6_6"/><span class="lab"><font color="Green" title="E65">A82</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">F</font></span>)</span>,<font color="Maroon" title="c12">f9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F9</font>,<font color="Maroon" title="c19">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">t19</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c20">t9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E50">A20</font></span></a></span>;<br/>
<a NAME="E15:58_1_1_6_6"/><span class="lab"><font color="Green" title="E66">A83</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c9">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58_1_1"><span class="lab"><font color="Green" title="E56">A28</font></span></a></span>;<br/>
<a NAME="E16:58_1_1_6_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A84</font></span>: 
<font color="Maroon" title="c21">t19</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c9">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:58_1_1_6_6"><span class="lab"><font color="Green" title="E64">A81</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E17:58_1_1_6_6"/><span class="lab"><font color="Green" title="E68">A85</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">F19</font>,<font color="Maroon" title="c19">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:58_1_1_6_6"><span class="lab"><font color="Green" title="E64">A81</font></span></a>, <a class="txt" href="#E15:58_1_1_6_6"><span class="lab"><font color="Green" title="E66">A83</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E18:58_1_1_6_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A86</font></span>: 
<font color="Maroon" title="c18">F19</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E19:58_1_1_6_6"/><span class="lab"><font color="Green" title="E70">A87</font></span>: 
<font color="Maroon" title="c19">F29</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:58_1_1_6_6"><span class="lab"><font color="Green" title="E68">A85</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E20:58_1_1_6_6"/>
<font color="Maroon" title="c17">F9</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c18">F19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:58_1_1_6_6"><span class="lab"><font color="Green" title="E63">A80</font></span></a>, <a class="ref" href="nattra_1.html#T32" target="_self" title="NATTRA_1:th.32">Th35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:58_1_1_6_6"/>
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<font color="Maroon" title="c11">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:58_1_1_6_6"><span class="lab"><font color="Green" title="E63">A80</font></span></a>, <a class="txt" href="#E14:58_1_1_6_6"><span class="lab"><font color="Green" title="E65">A82</font></span></a>, <a class="txt" href="#E16:58_1_1_6_6"><span class="lab"><font color="Green" title="E67">A84</font></span></a>, <a class="txt" href="#E18:58_1_1_6_6"><span class="lab"><font color="Green" title="E69">A86</font></span></a>, <a class="txt" href="#E19:58_1_1_6_6"><span class="lab"><font color="Green" title="E70">A87</font></span></a>, <a class="ref" href="nattra_1.html#T24" target="_self" title="NATTRA_1:th.24">Th26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c11">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> implies  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">g</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c12">g9</font> = <font color="Maroon" title="c11">g</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) ;<br/><span class="kw">assume </span><a NAME="E11:58_1_1_6"/><span class="lab"><font color="Green" title="E60">A88</font></span>: 
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">g</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c13">F1</font>, <font color="Maroon" title="c14">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c15">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font><span class="kw"> such that </span><br/><a NAME="E13:58_1_1_6"/><span class="lab"><font color="Green" title="E61">A89</font></span>: 
<font color="Maroon" title="c12">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E14:58_1_1_6"/>
<font color="Maroon" title="c13">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c14">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></span>;<br/><span class="lab"><font color="Green" title="E62">A90</font></span>: <font color="Maroon" title="c13">F1</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">g9</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:58_1_1_6"><span class="lab"><font color="Green" title="E61">A89</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c9">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="txt" href="#E11:58_1_1_6"><span class="lab"><font color="Green" title="E60">A88</font></span></a></span>
;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">t</font> = <font color="Maroon" title="c15">t</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c14">F2</font> ;<br/><a NAME="E17:58_1_1_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">g9</font>,<font color="Maroon" title="c10">s</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <span class="p3">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c9">F</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:58_1_1_6"><span class="lab"><font color="Green" title="E61">A89</font></span></a>, <a class="txt" href="#E15:58_1_1_6"><span class="lab"><font color="Green" title="E62">A90</font></span></a></span>;<br/><a NAME="E18:58_1_1_6"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">g9</font>,<font color="Maroon" title="c10">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1"><span class="lab"><font color="Green" title="E49">A19</font></span></a></span>;<br/><a NAME="E19:58_1_1_6"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">g9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58_1_1"><span class="lab"><font color="Green" title="E56">A28</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">F9</font>, <font color="Maroon" title="c18">F19</font>, <font color="Maroon" title="c19">F29</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c20">t9</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c17">F9</font>,<font color="Maroon" title="c18">F19</font>, <font color="Maroon" title="c21">t19</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c18">F19</font>,<font color="Maroon" title="c19">F29</font><span class="kw"> such that </span><br/><a NAME="E21:58_1_1_6"/><span class="lab"><font color="Green" title="E63">A91</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F9</font>,<font color="Maroon" title="c18">F19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c20">t9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E22:58_1_1_6"/><span class="lab"><font color="Green" title="E64">A92</font></span>: 
<font color="Maroon" title="c12">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">F19</font>,<font color="Maroon" title="c19">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c21">t19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E23:58_1_1_6"/><span class="lab"><font color="Green" title="E65">A93</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c12">g9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">F</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F9</font>,<font color="Maroon" title="c19">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">t19</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c20">t9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E50">A20</font></span></a></span>;<br/><a NAME="E24:58_1_1_6"/><span class="lab"><font color="Green" title="E66">A94</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c9">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58_1_1"><span class="lab"><font color="Green" title="E56">A28</font></span></a></span>;<br/><a NAME="E25:58_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A95</font></span>: 
<font color="Maroon" title="c20">t9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c9">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:58_1_1_6"><span class="lab"><font color="Green" title="E63">A91</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E26:58_1_1_6"/><span class="lab"><font color="Green" title="E68">A96</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F9</font>,<font color="Maroon" title="c18">F19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:58_1_1_6"><span class="lab"><font color="Green" title="E63">A91</font></span></a>, <a class="txt" href="#E24:58_1_1_6"><span class="lab"><font color="Green" title="E66">A94</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E27:58_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A97</font></span>: 
<font color="Maroon" title="c18">F19</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E28:58_1_1_6"/><span class="lab"><font color="Green" title="E70">A98</font></span>: 
<font color="Maroon" title="c17">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:58_1_1_6"><span class="lab"><font color="Green" title="E68">A96</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E29:58_1_1_6"/>
<font color="Maroon" title="c18">F19</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c19">F29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:58_1_1_6"><span class="lab"><font color="Green" title="E64">A92</font></span></a>, <a class="ref" href="nattra_1.html#T32" target="_self" title="NATTRA_1:th.32">Th35</a></span>;<br/><span class="kw">hence </span><a NAME="E30:58_1_1_6"/>
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">g</font>,<span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:58_1_1_6"><span class="lab"><font color="Green" title="E64">A92</font></span></a>, <a class="txt" href="#E23:58_1_1_6"><span class="lab"><font color="Green" title="E65">A93</font></span></a>, <a class="txt" href="#E25:58_1_1_6"><span class="lab"><font color="Green" title="E67">A95</font></span></a>, <a class="txt" href="#E27:58_1_1_6"><span class="lab"><font color="Green" title="E69">A97</font></span></a>, <a class="txt" href="#E28:58_1_1_6"><span class="lab"><font color="Green" title="E70">A98</font></span></a>, <a class="ref" href="nattra_1.html#T24" target="_self" title="NATTRA_1:th.24">Th26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">C</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span>)</span>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c6">i</font> #) as   <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a>  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D5" title="CAT_1:def.5">CAT_1:def 5</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c8">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) &amp; (  for <font color="Olive" title="b1">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font> holds <br/> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  ) ) &amp; (  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> ) &amp; (  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>  st <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> holds <br/> ex <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b6">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font> ex <font color="Olive" title="b7">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b6">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) ) &amp; (  for <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c8">C</font><br/>  for <font color="Olive" title="b2">F</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> holds <br/> <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F</font>,<font color="Olive" title="b2">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Olive" title="b2">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E18:58_1_1"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c8">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font> holds <br/> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  ) ) &amp; (  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> ) &amp; (  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>  st <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> holds <br/> ex <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b6">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font> ex <font color="Olive" title="b7">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b6">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) ) &amp; (  for <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c8">C</font><br/>  for <font color="Olive" title="b2">F</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> holds <br/> <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F</font>,<font color="Olive" title="b2">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Olive" title="b2">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E19:58_1_1"/>
 for <font color="Olive" title="b1">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font> holds <br/> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="txt" href="#E11:58_1_1"><span class="lab"><font color="Green" title="E54">A26</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> ) &amp; (  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>  st <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> holds <br/> ex <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b6">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font> ex <font color="Olive" title="b7">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b6">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) ) &amp; (  for <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c8">C</font><br/>  for <font color="Olive" title="b2">F</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> holds <br/> <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F</font>,<font color="Olive" title="b2">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Olive" title="b2">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E20:58_1_1"/>
 for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>  st <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> holds <br/> ex <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b6">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font> ex <font color="Olive" title="b7">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b6">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) ) &amp; (  for <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c8">C</font><br/>  for <font color="Olive" title="b2">F</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> holds <br/> <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F</font>,<font color="Olive" title="b2">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Olive" title="b2">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c9">f</font> implies <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1_1_7"/><span class="lab"><font color="Green" title="E58">A99</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c9">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">F19</font>, <font color="Maroon" title="c12">F29</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c13">t9</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c11">F19</font>,<font color="Maroon" title="c12">F29</font><span class="kw"> such that </span><br/><a NAME="E3:58_1_1_7"/><span class="lab"><font color="Green" title="E59">A100</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F19</font>,<font color="Maroon" title="c12">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c13">t9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:58_1_1_7"/>
<font color="Maroon" title="c11">F19</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c12">F29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">F1</font>, <font color="Maroon" title="c15">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c16">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c15">F2</font><span class="kw"> such that </span><br/><a NAME="E6:58_1_1_7"/><span class="lab"><font color="Green" title="E60">A101</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c15">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c16">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:58_1_1_7"/>
<font color="Maroon" title="c14">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c15">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></span>;<br/>
<span class="lab"><font color="Green" title="E61">A102</font></span>: <font color="Maroon" title="c15">F2</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c15">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c15">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c16">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c9">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E6:58_1_1_7"><span class="lab"><font color="Green" title="E60">A101</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F19</font>,<font color="Maroon" title="c12">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c13">t9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="txt" href="#E3:58_1_1_7"><span class="lab"><font color="Green" title="E59">A100</font></span></a>, <a class="txt" href="#E1:58_1_1_7"><span class="lab"><font color="Green" title="E58">A99</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F19</font>,<font color="Maroon" title="c12">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c11">F19</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">t9</font> = <font color="Maroon" title="c13">t9</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c15">F2</font>,<font color="Maroon" title="c12">F29</font> ;<br/>
<a NAME="E10:58_1_1_7"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c12">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">t9</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c16">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1_1_7"><span class="lab"><font color="Green" title="E60">A101</font></span></a>, <a class="txt" href="#E3:58_1_1_7"><span class="lab"><font color="Green" title="E59">A100</font></span></a>, <a class="txt" href="#E8:58_1_1_7"><span class="lab"><font color="Green" title="E61">A102</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:58_1_1_7"/>
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1"><span class="lab"><font color="Green" title="E49">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E21:58_1_1"/>
 for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>  st <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> holds <br/> ex <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b6">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font> ex <font color="Olive" title="b7">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b6">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c8">C</font><br/>  for <font color="Olive" title="b2">F</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> holds <br/> <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F</font>,<font color="Olive" title="b2">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Olive" title="b2">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> implies  ex <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b4">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font> ex <font color="Olive" title="b5">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font> st <br/>( <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b4">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1_1_8"/>
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b4">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font> ex <font color="Olive" title="b5">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font> st <br/>( <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b4">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<a NAME="E2:58_1_1_8"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b4">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font> ex <font color="Olive" title="b5">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font> st <br/>( <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b4">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E50">A20</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:58_1_1_8"/>
 ex <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b4">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font> ex <font color="Olive" title="b5">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font> st <br/>( <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c8">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b4">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c9">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c8">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> holds <br/> <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b1">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Olive" title="b1">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c10">F</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> implies  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c10">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c10">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>



<span class="kw">assume </span><a NAME="E22:58_1_1"/>
<font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c10">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c10">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<span class="kw">hence </span><a NAME="E23:58_1_1"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c10">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c10">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58_1_1"><span class="lab"><font color="Green" title="E56">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
